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IMSL_CHNNDFAC

# IMSL_CHNNDFAC

The IMSL_CHNNDFAC function solves a real symmetric non-negative definite system of linear equations Ax = b. Computes the solution to Ax = b given the Cholesky factor.

The factorization algorithm is based on the work of Healy (1968) and proceeds sequentially by columns. The i-th column is declared to be linearly dependent on the first i â 1 columns if:

where Îµ (specified in TOLERANCE) may be set. When a linear dependence is declared, all elements in the i-th row of R (column of L) are set to zero.

Modifications due to Farebrother and Berry (1974) and Barrett and Healy (1978) for checking for matrices that are not non-negative definite also are incorporated. The IMSL_CHNNDFAC procedure declares A to not be non-negative definite and issues an error message if either of the following conditions is satisfied:

Healyâs (1968) algorithm and the IMSL_CHNNDFAC procedure permit the matrices A and R to occupy the same storage. Barrett and Healy (1978) in their remark neglect this fact. The IMSL_CHNNDFAC procedure uses:

in condition 2 above to remedy this problem.

If an inverse of the matrix A is required and the matrix is not (numerically) positive definite, then the resulting inverse is a symmetric g2 inverse of A. For a matrix G to be a g2 inverse of a matrix A, G must satisfy conditions 1 and 2 for the Moore- Penrose inverse but generally fail conditions 3 and 4. The four conditions for G to be a Moore-Penrose inverse of A are as follows:

1. AGA = A
2. GAG = G
3. AG is symmetric
4. GA is symmetric

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